Current Status of One and Two-Dimensiond Numerical Modele: Successes and Limitations

by R. J. Schwartz, J. L. Gray, and M. S. Lundstrom Purdue University

School of Electrical Engineering West Lafayette, IN 47907

Abstract

The capabilities of one and two-dimensional numerical solar cell modeling programs (SCAP1D and SCAF'2D) are described. The occasions when a twedimensional model is required are discussed. The application of the models to design, analysis, and prediction are presented dong with a discussion of problem nreas for solar cell modeling.

Introdnet ion

Accurate numerical models for singie cryst,al silicon solar cells have provefi to be very reliable in the simulation of the performance of these cells. These models have proven to be exiremely useful in: the interpretation of experimental measurements; the identification of processes which limit cell performance; the prediction of benefits which will result from design and materials changes; the comparison of various cell designs; and the prediction of efficiencies which may everltually be obtained in silicon solar cells as various technological barriers are overcome.

The capabilities of a one-dimensional (SC.'LPlD) and a two-dimensional model (SCAF'2L) are described and examplm Df their use for each of the above r, rposes are given. I-' It will be shown that there are circumstances under which cells which appear to be onedimensional require a twedimensional model to properly simulate their behavior.

As cells become more efficient the requirements on the accuracy of the physics used in the model become more stringent. Effects which are of little significance in poor or moderately good cells can take on major significance in high efficiency cells. A number of problem areas which are of concern in t,he modeling of high efficiency cells are discussed. These include heavy doping effects, metal-semiconductor boundary conditions, minority car- rier mobilities, high injection lifetimes, and carrier-carrier scattering. Each of these may have a major impact on the performance of the cell under certain operating conditions.

The Model

Physical Effects of importance

One of the major advantages of a. numerical model is that it affords one the opportun- ity to include the very large number of physical effects which may be acting simultaneously within a solar cell. The cvmplexity of the phenomena and their interactions with each other preclude analytic solutions in anything except highly idealized situatioris, which are not indicative of actual cells or operating conditions. An attempt h a s been made in the formu- lation of SCAPlD and SCAPSD to include as many of the physical effects which are known to influence cell performance as possible and to do this in a manner which represents our

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present knowledge of these effects. One of the &oak in writing these codes was to have them be sufficiently accurate in their representations 59 that they could be wed in a predic- tive mode. This is possible only if p-11 of the pertipeut physica! effects are included.

In those cases where the physics is questionable, we have attempted to include options which allow one to choose between various models. For example, in the c s e of heavy dop- ing effects, one is able tc choose betxeen the models of Slotboom, Lanyon-Tuft, and Mahan, or to supply a subroutine of one's own choosing.

We have attempted to choose materials parameters which in our estimation are the most reliable. These materials parameters are used as default values. Tbe user can easily change these parameters to values that he views as more reasonable.

The following physical effects are included in the codes: hole and electron mobilities, including baeir doping and temperature de2endencies; heavy doping effects, using the for- mulation of Lundstrom, Schwartz, and Gray; absorption coefficients, including their tem- perature dependence; recombination, including Auger, Hall-Shockley-Read, and surface recombination. Surface recombination is handled through the specification of the surface recombination velocity. In the case of SCAP2D, the effects of surface potentials are also included.

Semiconductor Equations

The programs perform a full simultaneous numerical soluCion of the two continuity equations and Poisson's equation subject to the boundary conditions apprqriate to one and two-dimensional cells. The equations are formulated as shown in equations 1-3. !

V*J, = q(R-G). The generation term in equations 2 and 3 are given by

a3

G(x) = & cbae-""dX and the recombination term is given by equations 5, 6 and T.

733

(ND + -njA) rn =

- 1 +

The hole and electron current densities which appear in equations 2 and 3 are given by Nr

332

't (4) ' +i;

(7)

J, = -qpnnVv, + kTpnVn

A G vp = v-(l-q) - Q

(9)

where vp and v, are the effective potentials defined in equations 10 and 11 and A G and y are parameters which account for variations in the band structure, such as dcnsity of states m d band gap, and accoiint for Fermi-Dirac statistics.

No low injection assumptions are made. The equations are solved from contact to con- tact with approrriate boundary conditions so that the soiutions are valid for all ranges of operation and imlude minority and majority carrier flow. The latter d a c e ennie restric- tions on the CPU word size required for solution.

These codes have been extensively tested fw accuracy by comparing the results of their predictions with experimental results obtained on very carefully and extensivdy characterized cells for a wide range of cell designs and operating conditions. The ngeement has been such that a high degree of ccnfidence has been developed in results Smputed using these codes.

Code Descriptior,

Figure 1 is a block diagram of tlie Structuie of SCAPlD and SCAP2D. The operator must, supply information ahout the materials parameters, a description of the device to be analyzed, the type of analysis which he w+hes to perform, and t,he spectrum, if appropriate. He also can, if he wishes, control some of the details of the numerical solution; the amount of information supplied while the program is converging to an answer and how the output information will be stored or displayed.

The results of the ccimputation are presented in printed summary form and the detailed results of the calculation are storrd on magnctic tape. A separat,e plotting routine is used to access the information on tape and to display the appropriate parameters. The plotting capability is one of the most vzluable features of the code, in that it allows one to effectiveiy have a microscopic view of most of the parameters of interest in &;! interior of the cell under cperating conditions. W e will show some of the available graphical output as we discuss tho capabilities of t,he code. Table I show the input control offered to the operator. In every case default parameters are specified if the operator chooses not to sup- ply a parameter.

Table ll contains d lisding of plots which are available through the plott'ng program. In this case the operator specifics the type of plot which is required and the region of the cell for which he desires that plot. Most of the figures which follow were obtai7,ed directly from this plotti:ig routjne.

In addition to the reliability of the out.put,, the ut . i l i ty o f codes of this type will depend on their ease of use and efficiency of computat,ion. For ex.imple, in a design mode, it is

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c e l l Deesription

Type of Analpi9

--e

Print Results

---I Solve Tor

Choice n,p.\' and requested type

of analysis Materials - c

Parameters I

I I Optical Filters

Control Parametere

Control

~---{--l-[Cl on Tape for Future

I I-

Type of Plot end Range

Figure 1 Block Diagram of the Struc!ur? of SC'iWlD and SCAp2D

Table I - Input Parameters

Device Description Spectral Choices

Step junction AM 1.0 Erfc (N,,x,) Ex per imen tal Profile S W R E M II Uniform generation

Doping Profi!es A M 0

AM 1.5 direct & global Monochromatic

Dimensions User supplied

hiaterials Parameters Lifetime ( T and energy) Surface Recombination Auger Bandgap narrowing

Slot b oom Mahan L an yon-Tuft User suppLtA -.

Optical Filters & Refiection Filt.er (Ge, Si, Sic,, G A ) Back surface reflector

Types of Analysis Dark I-V Illurninat?d I-V Solar Cell Spectral Response

1

4 !

t

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Tsbfe II - Plotting Options

C,rricr concentrat ion Hole and e'rectron -:urrent :.mit ies Change iu potential (from .qii';tbriun) Doping density Energy band diagram Electric field Hole an.- electron quasi-electric field\ Erfective fields (elnctric plus quasi-rlrrt ric-)

Optical geaeration for holes and electrons

iole current density and components 5lectror. current density and components iicbility .ifet ime 3 at io of n,,/ni, %tent iai 3ecombica t ion rate ;'barge dcnqitv :scess carrier concertration 7

advantageous to he able to makc. n i ~ r l t i p l ~ i l ~ n ? ; in s renwnnhle length of time and at rea- sonable cost. Whiie SC':U,lD can he ri1ii die t i v d y on nearly s n y mainframe computer (a typical run on a CDC 6600 require.; 1GO-300 C ' I Y ' seconds). SC'.W2D requires a very fast machine with a large amount of w t i i n l nr virtual memory. On a i'yber 205. 300 CPU seconds are rcquired for a typical rut.

SitnatIuns Requiring Two-Dimensional Aidysis

In man! situations a onedimt~nsionnl siniulztion is quite sdequate and there is no need to use the more complex and esncnshc t\v*dimcn?;;iond simulation. On the other hand, there are a number of situations whit+ onl! a Iw*dimensionnl simrrlntion will suAice.

Some of the sit u3t ions which rcqitirc !wrdimcnsionnl annlysis are quite oh\-iour. while others appear o be onedirnrwion:If i n n n t i i r t b , hut. in f3c-t. reqirir~ 7 ttio-dimrnsiorx! s&i- tion for proper description of the cell ptirformnnce. Ifost of the C C ~ stiuc!ures which have been proposed 3s high eficiency silicon crlis fall into the olxic,iisly t\vedimcnsional analysis categorv. Among these structures nrr the In!crdigitnted Rnc-k Contact cell, the \ -crt i is l hiulti-Junction cell, the Etched Xlirltiple l - x t i c a l Junc.tir?n cell, t h e Polka Do: cell. and the Grating cell. -4s an example of the ase o f SC.QP2D in the analysis of these twedimensional cells, we shoh figures 2 through 4 for an IBC' cell. In Figure 2 we show the total shsrt cir- cuit current 3G.v under m e sun-condit ions. In Figures 3 and 2 we show the majority and minority carrier flow for this same cell operating under the same condition.

Less obvious appli<-ations of the two-dimeusional code are shown in Figires 5 through 6 in which a conventional solar cell h a s heen -.nalyzed. In Figure 5 we show the pot,ential distribution along the emit.ter from a poi:;: half way between the grid lines ui; to the grid lines u d c r open circuit conditions. This figure illustrates that there is a lateral voltage drop along the emitter, even under open circuit conditions, as a result of the current which is injected in the vicinity of the grid line. Figure 6 shows the circ~iating currents which exists in the vicinity of the grid k-.

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Figure 2 Total Short Circuit Current. for an IBC C4I I Sun AhlI.0

n+cmtat t 3+ conwt

f i

0 L U 4

v E

3

Figure 3

!

x (microns)

Alajority C'arricr Current Flow for t h p ('ell of Figure 2

336

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Figure 4 !.linority Carrier Current Flow for the Cell of Figure 2 L.

- 4

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Figure 5 Potential r)istrihut,im in thr Emitter 01 a Conventional Solar Cell Operating at 900 suns. 1

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- 7- 3:

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-.

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.

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Figure 6 Circulating Current in the l'icinity of a Grid Line for a Conventional Solar Cell

An w e n !ess obvious problem with one-dimensimal simuhtion occurs when one tries to properly nodel the front surface of a conventional cell. This surface is composed of a SiOTSi iHterface anti a metal-semiconductor contact. In a one-dirneosimal simuiation, one is forced to aggregate the two efiects with some equivaknt front surface recombination v e l e city, SF. Figure 7 illustrates the difliculty with this approarh. Under short-circuit condi- t.ions the proper value of SF is equal tG the surface recombination velot-ity of the S;O,Si interface. However, near oppn circuit conditicns, t b ~ propcr v: . sf SF may be 3 to 4 ord- ers of magnitu4e larger. This is a result oi the fact ths: the . ! I serniconductw contzct. may be a very effective recombination site for minority czrrlers. . ;s particularly important as the operating voltage of the cell increases. For proper oper.!tion of a onedimensionaI code, the front surface recombination velocity a i i~uld be a function of operating condition. The twedimensional code does not have this problem, since the snrfzce recombination v e l e city at the SiOTSi interface and the metal semiconductor interface are specified separate!y , and the reconloination along the entire surface is properly accounted for under 311 operating condi tioas -

At high operatitig conditions, such as arc found in concentrator solar cells, even the conventional cell behaves in a two-dimrnsional fashion and must be modeled using the twedimensional code. Xlinority carrier current flow for a cc.nventional cell operating at 800 suns is shown in Figure 8. If this cell is modeled using the one-dimensional code under these operating conditions, swious errors are encountered in the cmi>tltat,ion of the fill fac- tor which can not be compensated for by including an extcrital serias resistance in the model, as the effect is nonlinear.

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loa l& u)

10' 1

1

10' 1- - . - - - - - -0.10 0.09 0-IC 6.20 0.30 0.90 0.50 0 . 6 0 0 .

Junction voltage (volts) 0

Figure 7 Efiective Surface Recombiaation Velocity as a Furxtion of Operating Voitage and the SiOrSi Surface Recombination Ve1ocit.y L

Figure 8 Minority Carrier Current Flow for a Conventional Solar Cell Gperatir ~ at 800 Suns (V=.soO volts 5 ~ 2 1 . 6 amp/cm2)

339

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Modes of Utilization

As we mentioned previously, a carefully prepared solar ce!l model is useful in a number of modes. In this section we will discuss the use of SChplL and SCAP2D 9s a design tool, a sensitivity Analysis tool, an aid in the analysis of experimental data, an aid in tbe provi- sion of insight into the operation of the cell, and, finally, as 3 predictive tool for the com- parison of pro- cell dcstgns and as a means of projecting performance as various tech- nological barriers are rerwved. For the sake of continuity. we have chosen to use the San- dia nigh concentration cetl operating at 1 sun as a base line design. This is a cell which has exhibited 18% catversion efficiency rt one sun, and 20% conversion efficiency in the .5r)-lOO sun range for an AM 1.0 spectrurn.

Design

As a simple example we show, in Table ID, the effects of variations in the base dophg about the present design doping of 2.29 x IOi6, on the performance of this cell. We see tha t the present base doping is nearly optimum for the desigrl parameters used in the other parts of the cell.

Table Ill

Solar Cell Performance Dependence on Base Doping AM 1.0 (one sun)

Base Doring cm-

s x 1015 1 x 1ol6 2-29 x 1016 1 x lo*' 5 x 10''

Sensitivity Analysis

L c VGIts

-634 .a0 .649 -656 -650

J c maJern2

35.1 34.8 34.4 33.3 30.2

F.F.

.828

.833

.EM

.838 -836

Efficiency %

18.35 18.46 18.55 18.21 18.37

By utilizing a computer code such as SLFREM io simulate fabrication conditions one can model the sensitivity of device performance to fabrication parameters. Here, as an extreme case, we examine the effects of changes in the emitter doping prcfile on cell perfor- mance. The Sandia cell was simulated using the two emitter profiles shown in Figure 9. In Table lV, a comparisort of these simulations is shown. Note that the erfc emitter profile simulation predicts a higher VoC. This is due to the lower net recombination in the emitter as compared to tbe SCiPREM JI emitter profile simulation, as shown in Figure 10. Recom biiiation is higher in the SlJPREM II emitter because the doping is higher over most of the emitter volume, and therefare Auger recombination is correspondingly higher also.

If the results of 8 process-sirnu!ation program such as SUPREM are coupled with SCAPlD 9r SC.4P2D as shown above the sensitik-ity of the cell to process variations can be readily eszablis h en.

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Figure 9 Emitter Doping Profiles as Determined by SUPREM II and Complimentary Error Function

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tY .c - - - - - - - - w a- '%;: u a QZ

I- .loo0 .5m 2 - c t-

LL Q

.2% 5 !

+ I I t \

H

c

L L 3 .ooo

.MU .%OO .Em .ROO 1 .Wil D I STfiNCE I N M !ZROI\IS

Figure 10 Emitter Recombination for the Two Doping Profiles Shown in Figure 9

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Table IV

Dependence of cell performance on emitter doping profile AM 1.0 (one sun)

Type of Profile V J c FF Efficiency V O L 1*,a/crnt %

Erfc .648 34.3 .836 18.6 SUPREM II .632 33.9 -833 17.75

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It is possible, by adjusting the parameters entered into the code, to obtain a fit between the model generated results and experimental dark I-V, solar cell, and spectral response curves. if this fit can be obtained for a siagle set of parameters, then one has a r emnab le expectation that these are the correct parameters describing this device.

With the ability to observe most of the parameters of interest as a function c' pmition and operating I -3itions anywhere within the cell, it is possible to achieve a grtr. ded of insight into the :miting factors on any cell design. Examination of the wodel for.the 205; Sandia cell very quickly establishes that the cell appears to be emit.ter limited, and, in fL, ., tbat further efforts in imrroving the performance of the cell should be devoted to reduction of the metal-semicondur tor contact recombination and in reducing the volume of the beavily doped emitter.

Prediction

Potentially one of the most valuable, and also one of the most risky, uses of the numerica! models is as a predictive tool. The models have d:eady been shown to be quite reliable in comparing the relative merits of different cell designs. One particularly attrac- tive way to utilize the code is to use it to identify limit,ing phenonienon in a particular cell design and then to remove that limitation and cbserve the effect on cell performance. In this fashion, one can predict benefits which will accrue through various advances in technol- ogy, and, in fact, can make some rensonahle estimates of the ultimate performance of sili- con single crystal solar cells. This latter use of t,he code is particularly risky since as the performance of the cell improves, physical effects which may have been insignificant in their effect on cell performranee before, may suddenly become the dominant limitation.

Problem Areas

There are a number of areas in which there is concern about existing solar cell models either because the physics is not me11 understood, available data is thought to be unreliable, or because the eflect has iiot been include in the model. These areas of concern are dis- cused Oelow.

Heavy Doping Effects

There is a controversy over the origins and magnitude of heavy doping effects. There

very heavily doped samples where we have our major concern. In order to alleviate this situation somewhat, we have provided the operator with the option to choose between most of the popu!ar band gap narrowing models. This remains an area of major concern and is probably the least reliable area in the modeling of silicon solar cells.

Anger Recombination

m is a great deal of scatter in the measured effective band gap narrcwing, particularly in the .I

1 1

Some uncertainty exists about the reliability of published Auger coefficients. At least two groups (Sandia and General Electric) have indicated that published Auger coetkicienta may be too large.

Minority Carrier ?. iobility

Reliable measurements of minority carrier mobility do not exist. Various authors have proposed that the minority carrier mobility is larger, smaller, and the same as the majority csrrier mobilities of the same type carrier. As a consequence, SCAF'lD and SCAP2D assume that the minority carrier mobilities for electrons are the same as they would be if electrons were majority carriers. A similar assumption is made for holes.

Metal-Semiconductor Contacts i

In well designed high efficiency solar cells, the metal semiconductor contact limits the open-circuit voltage. The removal of this high dark current source, through the use of tun- neling contacts or through the reduction of the metal-semiconductor contact area, has already demonstrated a significant improvement in open-circuit Toltage. Further advances in this area may well employ heterojunction structures in additim to the present tunneling structures. SCAPlD and SCAPSD allow for specification of a finite minority carrier surface recombination velocity to model this effect.

Doping Proales

We have already seen that device prrfcrmance can be a strong function of the shape of the emitter doping profile. SCAPlD 7,nd SCAPSD allow for the use of a complimentary error function, a computed profile based on tb: Fair diffusion model for phosphorus, doping profiles obtained From a process simulation program such as SUPKEM, or experimental data. The use of data from SIMS measurements has the problem that it includes the total impurity concentratim not just the elec!.rically active dopants. If any precipitation is present in the highly doped region, SIhlS will overestimate the amount of active dopant. Spreading resistance measurements are a measure of the free carrier concentration. Near the depletion region this can lead to significant errors in the cloying profile if the spreading resistance profile is interpretated as being the same as the doping profile.

f

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Spectral Response

Spectral response measurements are particularly useful for obtaining information about the base lifetime and the surface recombination velocity. However, some difficulty is experienced in matching long wavelength response with computed response. This difficulty has been traced to the fact that small changes in device temperature can lead to large changes in the long wavelength response as a result of changes in the absorption coefficients due to a shift in the band edge.

In order the determine the surface recombination velocity cf the Si02-Si i n tdace , it is also highly desirable to have spectral response measurements in the very high absorption regime of .35 - .4 pm. Accurate measurements of the internal quantum efficiencies arc disticult to obtain at these wavelengths.

Effects oP Band Gap Narrowing on Long Wave Length Absorption Coemcients

At the present time no corrections for the effect of band gap narrowing are made to the absorption coefficients.

Carrier-Carrier Scattering

Carrier-carrier scattering can bc a significant effect in high concentration solar cells, and will become a significant effect in one sun solar cells as the efficiency is increased.

High Xn)ection Lifetime

At the present time very little data is available on majority carrier lifetime. A typical modeling approach is to assume that the majority carrier lifetime is the same as minority carrier lifetime. This seems to give ressonably good agreement with cell performancr under high iujction conditions, but direct measurement of the high injection lifetime would be highly desirable.

Conelnaiom

One and twedirnensional device models have been quite successfully employed as an aid to design, interpretation, sensitivity analysis, and prediction. However, the predictive capability of any device code is only as good as the physics which is modeled and the data which is supplied. If further improvements are to be made in the performance of single cry- stal silicon solar cells, careful attention will have to be paid to both ol these areas and a great deal of effort will have to be devoted to measurement techniques which will allow the independent determination of the parameteis which must be supplied to the device code.

Acknowledgment

SCAPlD and SCAPSD were developed under the sponsorship of Saudh Natioval La- boratories on contract number 52-567.5.

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M. S. Lundstrom, R. J. Schwarts, “Annual Re rt on Interdigitated Back Contact Solar Cells,” TR-EE 80-14, School of Electrical r ngiaeering, Purdue University, West Lafayette, IN. R. J. Scbwartz, M. S. Lundstrom, J. L. Gray, “Annuttl Report on High Jitensity Solar Cells, TR-LE 82-5, School of Electrical Engineering, Purdue University, West Lafay- ette, IN. R. J. Schwsrtz, J. L. Gray, M. S. Lundstrom, “Report on High Intensity Solar Cells,” TR-EE 83-21, School of Electrical Engineering, Purdue University, West Lafayette, IN.

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DISCUSSION

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SAii: I m u l d like to ask you about that particular example you save, the ~011- plmntary error function, also the SUPsgLl. in the results. What is the basic mechanism? What are the recambination mechanisms t%at give you the difference?

There is quite a difference

S-2: Mat's included in the code is Hall-Shockleg-Read and Auger using Schmidt Auser coefficients, and for that particular run -- thank you for asking -- I should have mentioned at the time: €or that particular run we used the Slot-Bo- band-sap narrowing model, so there is a significant amount of band-gap narrowing occurring, and recombination :echanim. For that particular run, Auger and Hall-Shockley-Read, and I didn't bring the plot along. I don't recall ehat the split wns.

The plot shows the split between them.

SAH: For the particular profile, was ihe Auger that causes the one to be better than the other me?

SCWARTZ: Yes, I believe that it wad Auger, but I don't have the plot with m e .

MEUGBOSCHBL: You said that the published Auger cciefficients don't agree with In order t o get agreement, do you need larger lifetimes the experilasat.

or shorter lifetimes?

SCHVARTZ: The recombination wants to be reduced. I should qualify that a little bit. I was repeating & a t is said in a couple of publications by Posene at GE and Weaver at Sandia. It is possible that the problem lies in the band-Eap narrowiog model and not in Auger, so one wants to be a littlc careful. There is a problem in the emitter, and that's clear, and people have tended to blame Auger. Chat it is Auger -- it amy dell be related more to the band-gap narrowing.

I gueas 'f'm not completely convinced

DAW: I would like to follow up on Sah's queetion. You have the same carrier density at the surface for the SUPZZH and for error function. So, norm- ally you have much larger field right ai: the surface in case of comple- mentary function, and I would expect less recombination there. Would you give some reason why?

SCHUARTZ: Yes. There are a nUmb8r of reasons. One is that the recombination v only depends on the Auger coefficients and lifetime, but also on the c...cess minorit;- carrier concentration, and if I had shown the plot, whet you would t.ave seen is that *any of the ccieriers are recombining in the case of SUPREM as they moved, and the axis cure was lower at the surface. The othar diffecence is in the way that the band-gao narrowing effective field is distributed. The minority carriers in the emitter don't see just an electrostatic driving force due to the gradient. There is another component, which is associated with band-gap narrowing itself, and it tends to reduce the effect of pulling minority .,aerier8 away from that surface or keeping them out of the emitter. are distributed differently.

Both ace operating and they

m -- --

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1 DAUD: The secotd question has to do with actually running this program. We find that one of the items that one has to put in is the 'I Trio. Wora&lly when we measure, we either matmre the life ime or diffusion length where the doping is already there. H o w does one reconcile with this? M a t kind of numbers one should put for 'I@

%no?

Po and

SCHWWlZ: I'm sorry. You say you normally measure what?

DAUD: Soy we hav-2 a cell where we measure the diffusion length or the life- time. corrects it for the doping.

Ue cannot directly put that in your model because ywr model

SCKVARTZ: Yes. There were a lot of features that were not talked about here, and since we have sent a copy of the code out to JPL, he is asking. You have the option of turning on what amounts to a Kandel fit to doping. If in fact you have that turned on and if the base doping is above the transition doping foi the the Kandel fit, then you have to correct that. If you have measured the lifetime at that particular doping level, you either have 50 correct it or just turn Kandel fit off and snter the value you masure. It's your choice. It is under your control on the input deck.

Lf#DHOLLI: I have three questions. They are all, I think, fairly quick. Just to remind the audience: you recast soate of the -- what you might say, equations which xere truer to the physics -- into a form that one is more used to seeing in a conventional treatment of semiconductor device physics. In so doing, you introduce the parameter that you called small gamma, lower case gamma, and I think that that parameter war supposed to have taken care of various things that were being violated by the density of states. that the parameters that go into the model have to be measurable. So with that preparation €or the audieilce, I could have just asked you what success havb you had in twasurinb gama, and how do you do it? you do it, if you cap't do it?

You made a big poiit, which is an extremely valid p i n t --

How don't

SCWARTZ: I tried t o stay aGay from the equations, Fred, and I apologize for The gama that Fred is talking about entered into

Here is the electrostatic poten- the poor quality here. the effective potential that we showed. tial, and here is the term hich I said was an effective ar'xetry factor, and that term is not normally known. effective band-gap narrowi.Ig, which in fact looks like this -- the band gap plus all the degeneracy and Land structure effects. Gama, in fact, has electron affinity divided by the delta G minus terms for degeneracy. The answer to your question is, you don't know. But before I let that go, it. turns out that for solar cells you seldom care, end the reason fc? that is the following: GB observed this first, the range on that nvpbers fros 0 to 1 and one can run the fuli range and see almost no detet. Able change i n the device characteristics. On the other hand, if you L o k inside the device, there are radical differences in the electric field distribution ? n the emitter, in that region. There a:e huge differences. But it turns out, and you can do this in closed form, that i f you are dealing with a region which is quasi--neutral and low injection, ao the

As one gets measurtlaent of the

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emitter is under anything except extreme cmditions, you can show i n analytic form that, in fact, terminai characteristic is independent of that parameter. There are places where it makes a difference. make a difference in junction capacitance: you can shon that fairly readily. It makes a difference ia the electric field. And a dramatic one i f , for instance, you ar- eonzerwd with avalmche, not in a solar cell, but calculations of avalanche, and yot are in a heavily doped condition. Then possibly OR? shouLd be worried about preseyt-day avalanche coefficients if they are Sased on classically calculated electric fields. But the answer is that it doesn't affect, the terminal characteristics.

It does

LIlDHOtW: The r e a ~ t n I asked tirat is twofold. First of all it would seem to me that for diagnostic purposes it would be desirahle to h o w what you call gaema, or what I would rather just call the eiectrct. 'finity, and I know about the coamrent you made about tie quasi-neutral on. you start entering a little bit into thc junction transition :egion, then I think it becomes more important. And thc reason i asked that is, if i correctly read your earlier paper with Mark Lundstron, that you indicated a method for masuricg gaarma. And so I come back to mv o-iginal question: what degree of success have ynu had in measurinf garwa?

But as

SCHWAPTZ: Very little. We're still working on it .

t1M)HOLM: ? think it is 'a gaod thing to work on, actualiy. The reason is that it's very 8881, even though you did not intend to do this -- in fact, your wording was ver: careful -- but people will take sort of :- :!a1 cases and say it doesn't matter, but it does matter from a diagnosti; point of view in finding out whet's going on with the profiles,how you can improve the device, that kind of thing. I ' m very glad you used such careful wording, so congratulations on that.

SCHWABTZ: I'm glad you read the paper, b.ry few people have.

LIIUDHOW: Its a very interesting and very good paper. I was extremely int,r- ested in your measurement of gamma.

SCHWABTZ: I have a Ph.D. student who is extreuoly inter sted tro.

LIBDHOLU: The other thing that I aoted that you said, an? since you word things so carefully, I was noticing that you said %hat most of the p e o p L who made electrical measurementa in effect were measuring the p-n prod- uct. Mow, I think that that's probably true of the Slot-Boom graph you UBI$, the transistor structure; I think It's not true of ERIC peoplz. Would you agree with that? EBIC data and the guys fro= GI3 who aren't here pre talking as. A;'.. p-n product and I think that they can't do that. The fundamental reason is that Fermi levels have to remain sensibly, spatially anvarian.; c*-jr a significant region of the device in order fcir that measurement to yield the p n product.

A good portion of the data now c,=2..:& out is

SCHWAElTZ: That's absolutely right, and from a physicG point of view is ver:- pleasing. And from A modeling point of view it's difficult. because now

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in tho intermediate region. where one worries about what happens to the brad rhrpe, you've got a problem in using it.

LT?iDtiOUl: On0 last one. You keep mentioning twodimensional programs, and I rrculd like to know about three-dimensional programs, and uhy you CM get a m y with two-dimnsional .

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SClNILpn: It's not so much Eetting nuof with two-diwrrioarl, it's really a level of difficulty. U8ghCted it inadvertently - - L told you that It cost about $5 .00 and take& 100 to 300 seconds on a 6500 CDC if you run the 1-0 code. about 300 CW/second OE our CIBBP 235 Supercmputer to run the 2-0 code. low, what I m a a 1 by that is to m u a full analysis. something like 10 points, m d do 611 the associated etafE. That uses a h u t 2,000 mesh points. 30 right now it's about what ue rre capable Lf tackling. If Putdue will put another two million b*otds i n t o the main wrory of t h a t computet then c;e hill lo<llt at the three-dimensional.

In our t-se -- one thing i didn't -ntion, and T

It takes

LIKD!lOLM: If the next SpeAket is successfil in catting the Computet time -.i&ui€icantly by h i s techniqve, as he suggests, would you then strongly advocete three-dimensional stabdy-state mdeliag as a highly useful, moderately useful c: raL.ely useful vehicle for solar tell design optiri- zatioo and for IOV '--?zing manufacturing processes?

SCWARTZ: I think tho;.. a very good questiun, whether it's asked about one, tua x three-dirensimal, gnu the anshr'- l i e s in how easy it is to use, whetbet it 's € u t , are the ta - -a tounds quick, .nd what is reasonably 4e.p. Because bou do have to nake a ll-t of runs, and if it's very eqsasive or -:sry time-consuming, +he utility becomes a lot less.

LlYDW)LLI: Suppose it doesn't cost arlykhing?

S C h . m : Then its very usefrtl to do oven for one sun. Io that what you think?

L1lMK)LLI: I heveJ*t thought enough cbout it.

S W A R T Z . If you nant to do a cell like Dick Swansor; :.

LINDHOUI: A more coaveational cell is very useful there? O r is it moderately uceful:

S W A R T l . : I daubL, it, I can't see t h e benefit to a conventinnal cell with th:ee dieensiouu.

QUKSTIOY: Dick, a quick yirestion. Did you decide t h t radiative rhcombina- tion and trap Augsr effect could be neglected safely?

SCWARTZ: PI, we didn't. I told you the status of the code as it w o o . It is c fa:rLy straightforward ratter to add tbosa components to it, we jast haven't drivca tt,i.rga, we haven't aac'n any :Ins where the other lifetime- li!r,;ltia& mett 3as were low enough to dc that. But. clear-,, if JOU 2 3 , t h a t ' s a limi.ii16 mechanism that is noc present end needs co be added.

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QUESTION: And or, the surface recombination you Right also have trap huger effects. or do you wrap it a11 up in 8 surface recabination velocity?

SCWAR'YZ: -11 in the one-direusionel code it's wrapped up in the surface re- combination velocity. dimensions. In two dimusions you could either do it by LriteLrating through the trap states and capture cross-sections. or you could do it by a lumped parameter. which is probably not as good - - and you do have t o control surface potential. whicb we do by settin& t h e c h a r ~ e in the oxide.

I didn't talk about hod it's handled in two

QURS'IIOY: W o w , a last question: capacitance. Do you work it out or do for: do current-?oitage. cqacitance voltage?

SCHUARl2: Yo. JPL doesn't know if their version of the code does have capac- itanc, in it. re just didn't tell ther. It is the quivalent of very low-frequency capac;tance 1.11 tell you what it is and you can name it. priate voltage term put in, and you are quite right.

It is the ina.cgzal of either the electrons or holes with the appro-

QUESTION: It can miss by a factor of three or four?

SCliWARIZ: Yes. We don't use 't that May.

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